This paper presents an interpolation-based method (IBM) for approximating some trigonometric functions or their integrals as well. It provides two-sided bounds for each function, which also achieves much better approximation effects than those of prevailing methods. In principle, the IBM can be applied for bounding more bounded smooth functions and their integrals as well, and its applications include approximating the integral of sin(x)/x function and improving the famous square root inequalities.